Channel Capacity Optimization Using Multiple-Input Multiple-Output for a Rayleigh Channel

The use of multiple-input multiple-output (MIMO) systems with multiple antenna elements provides an efficient solution for future wireless communication. In recent times, wireless communication has been characterized by high speed, higher data throughput, and high Bit Error Rate (BER) in a limited bandwidth. Low data speed handover, low Signal Noise Ratio (SNR), and hardware power drainage are common channel problems with wireless systems. The objective of this research seeks a way to optimize channel capacity and enhance system performance. The paper presents a comprehensive performance analysis of channel capacity under the Rayleigh fading channel using water water-filling technique. MATLAB was used to analyze and simulate the process. Simulation results revealed that the water-filling algorithm can effectively optimize channel capacity in the wireless communication system.

Introduction

Today wireless communication system is faced with several channel impairments including but not limited to low bit error rates, drop calls, high power consumption such as those experienced in the third generation (3G), Long Term Evolution (LTE), worldwide interoperability for microwave access (WiMAX), and wireless fidelity (WIFI). To overcome these challenges, the MIMO communication technique was introduced to provide high quality of service, increase the data rate, improve coverage, reduce outage probabilities, and carry out another performance matrix of data communications. MIMO guaranteed spatial diversity gains to overcome small-scale fading effects by using diversity. It reduces path loss, by using array gain in the distribution of the antennas. It increases capacity by spatial multiplexing using parallel stream transmission and Spatial Division Multiple Access (SDMA); by focusing on a narrow transmission beam using a smart antenna. MIMO uses beamforming to enhance interference reduction and spatial multiplexing, and special MIMO for close-loop spatial multiplexing [1]. Orthogonal frequency division multiplexing is a popular wireless multicarrier transmission technique. The basic principle of OFDM is to split a high data rate stream into several low data rate streams so that the lower data rate can be transmitted simultaneously over a number of subcarriers (Spatial Multiplexing) as asserted by [2] and [3]. According to [4] and [5], different signals in MIMO systems are transmitted orthogonally at different positions in space and at least one position should be in a free fading dip.

The transmission energy can be managed and minimized with an optimum resource allocation technique. This resource allocation is fundamentally based on the convex optimization problem which uses the water-filling algorithms principle [6]. In his study, [7] concluded that more power is allocated to a sub-channel with better SNR, to maximize the sum of data rates in all sub-channels, according to Shannon’s Gaussian capacity formula where each sub-channel’s data rate is related to the power allocation. Each used sub-channel’s SNR information will be fed back to the transmitter so that the water level is derived from the fading channel coefficient and the noise power.

The main objective of this work is to optimize the channel capacity of wireless communication systems. To achieve this, an iterative Water-filling algorithm is used for the MIMO system. The performance analysis is recorded based on the simulation results of ergodic capacity, Bit-Error-Rate (BER), outage probability, and channel gains using Binary phase shift keying (BPSK) modulation techniques. MATLAB simulation was used during the simulation process.

The rest of the paper is organized as follows: related literatures were discussed in Section 2. The research methodologies and system model were defined in Section 3. In Section 4, results and a discussion of simulation were presented. Finally, the summary and conclusion are presented in Section 5.

Related Literature

A MIMO system was inspired by papers presented by two renowned engineer scientists; Telatar and Foschini. They inferred that there is a dramatic linear increase in channel capacity with some antennas [5]–[7]. In [8], common practices of capacity enhancements were in the improvement that diversity projected in signal transmission, both in time and spatial diversity. Common assumptions in [9] consider two main factors, time division duplexing (TDD) mode, and hardware capability when examining the required processing performance within the coherence time interval in capacity enhancement. However, MIMO systems allow us to operate two distinct dimensions of a radio link; the first being the Diversity and the second being the Capacity [7]. In [7], [10], and [11], channel capacity increases as the number of antennas is increased, proportional to the smaller of the number of transmit antennas and the number of receive antennas. Signal fading between pairs of transmit and receive antenna elements are independent and identically shared. Reference [7] inferred that MIMO systems capacity can be linearly increased with the number of antennas. This according to [9] is due to its aid in diversity combining which increases SNR, thus resulting in capacity increase.

Spatial Diversity is employed to make the transmission more robust, though it does not increase data rate as inferred by studies carried out by [12] and [13]. Spatial diversity can be in receiving diversity and transmitting diversity. See illustration in Figs. 1 and 2.

Fig. 1. SIMO configuration.

Fig. 2. MISO configurations.

In [14], an effort was made to address the pre-coding problem in MIMO ad hoc networks by maximizing the system’s mutual information under power constraints. A fast and distributed algorithm based on the quasi-Newton method was developed to solve the optimization problem. Several studies [15]–[18] on beamforming techniques for MIMO capacity improvement of wireless communications provide a significant survey on the evolution and advancements in antenna beamforming and the setting of different requirements for indoor and outdoor communication scenarios and announce some basic concepts of beamforming. Scheduling is a traditional concept of simple “orthogonal” time division multiple access [16] which was enriched through the ideas of random access known as ALOHA [19]–[22], which studied a scheduling algorithm for minimizing the packet error probability in clustered TDMA networks. Their studies for a clustered wireless network prove that it may be used to improve channel capability. In another study, [23], it was discovered that transmissions from more than a single user cannot co-exist successfully in the same time slot, therefore assumed and recommended Time Division Duplexing (TDD) for data scheduling as a duplexing mode as a key enabler for a new heterogeneous network architecture with the potential to provide ubiquitous coverage and unprecedented spectral area efficiencies. The study proposed the distributed implementation of a weighted sum of rate maximization-based power allocation using a dual decomposition approach. Bensky [24], noted that power allocation plays a significant role in deciding MIMO capacity unlike in SISO. The fundamental question is the magnitude of power allocated to each transmit antenna.

Power allocation is an optimization problem to maximize capacity. Depending on the closed or opened loop mode, allocation is based on CSI. Intuitively we will allocate more power to channels based on performance quality.

Ideally, channels are expected to be naturally very turbulent due to physical and logical phenomena along and within which they are composed. The question of whether all the approaches that were studied in the literature as means of optimizing the channel affect its performances in resolving data distortion/contamination in wireless communication has to be addressed. In this research, we therefore analyzed the water-filling algorithm by means of considering the channel SNR and making an assumption that $\sum p_i=p_T$, which is a constraint on the channel. This is called the Constraint of Optimization problem. This optimization problem was resolved using the water-filling algorithm. As an analogy to water in a pipe filling a tank, this algorithm distributes signal power depending on the presence of noise in the channel. More power will be distributed to channels with less noise. This optimum power allocation aids power optimization and enhances data throughput in the wireless communication channels.

Research Methodology

Wireless Communication System Model

The system model of a wireless communication system is, to consider a transmitter with N transmits antennas, and a receiver with M receives antennas. The channel can be modeled by the $\text{M}\times \text{N}$ matrix of H. The channel can be represented by the N × M matrix of the H channel. The N × 1 received signal y is presented in Fig. 3, given as

(3.1) $$\begin{array}{r}y=Hx+n\end{array}$$

where x is the M × 1 transmitted vector and n is the N × 1 Additive White Guassian Noise (AWGN) vector, normalized so that its covariance matrix is an identity matrix.

(3.2) $$\begin{array}{r}\text{Rss}=\frac{{\text{E}}_{\text{s}}}{{\text{N}}_{\text{r}}}{\text{I}}_{\text{N}}\end{array}$$

Fig. 3. MIMO channel model.

$E_s/N_r$ is equivalence to the power of each antenna and I_N, the identity matrix of N. In the formula, the H can be defined as

(3.3) $$\begin{array}{r}H=\left[\begin{array}{ccc}{h}_{11}& \cdots & h_{N1}\\ ⋮& \ddots & ⋮\\ h_{N1}& \cdots & h_{NM}\end{array}\right]\end{array}$$

MIMO Channel Capacity

The capacity of a random MIMO channel with power constraint P_T can be presented as an expected value of

(3.4) $$\begin{array}{r}C=E_H\left\{\underset{p\left(x\right):tr(\mathrm{\Phi })\le P_t}{max}I(x;y)\right\},\text{ bps}/\text{Hz}\end{array}$$

where Φ = E${\left\{x{x}^H\right\}}^2$ is the covariance matrix of the transmit signal vector x. Irrespective of the number of transmit antennas, the total transmit power is limited to P_T.

Estimating the capacity of a MIMO system the channel state information (CSI) is a critical parameter that must be determined.

Case 1:

When the transmitter does not know the channel; In this case, it is optimal to use a uniform power distribution [19]. Its ransmit covariance matrix is represented by

(3.5) $$\begin{array}{r}\mathrm{\Phi }=\frac{P_T}{n_T}{I}_{nT}\end{array}$$

Uncorrelated noise can also be assumed in each receiver branch which is described by the covariance matrix given by

(3.6) $$\begin{array}{r}{K}^n={\sigma }^2{I}_{nR.}\end{array}$$

The mean capacity for a MIMO channel can therefore be presented as:

(3.7) $$\begin{array}{r}\text{C}=E_H\{{\log }_2[det({\text{I}}_{\text{nR}}+\frac{\mathrm{\rho }}{\text{nT}}{\text{HH}}^{\text{H}})]\}\text{bps}/\text{Hz}\end{array}$$

where ρ = $P_T/{\sigma }^2$ is average signal-to-noise ratio (SNR) at each branch of the receiver. But by law of large numbers, the term $(1/nT)H{H}^H$→$I_{nR}$ as $n_T$ get larger and $n_T$ is fixed.

MIMO channel ‘H’ can further be analyzed by taking diagonal the product matrix $H{H}^H$ either by eigenvalue decomposition or by singular value decomposition. Using eigenvalue decomposition, the product matrix is given as

(3.8) $$\begin{array}{r}H{H}^H=E\wedge E^H\end{array}$$

where E is the eigenvector matrix with orthogonal column and ∧ is the diagonal matrix with the eigenvalue on the main diagonal. Using this notation, the mean channel capacity will be represented as:

(3.9) $$\begin{array}{r}\text{C}=E_H\{{\log }_2[det({\text{I}}_{\text{nR}}+\frac{\mathrm{\rho }}{\text{nT}}\text{E}\wedge E^{\text{H}})]\}\text{bps}/\text{Hz }\end{array}$$

By using the Singular value decomposition the product $H{H}^H$ on the channel matrix H is given as:

(3.10) $$\begin{array}{r}\text{H}=U\mathrm{\Sigma }{V}^H\end{array}$$

U and V are unitary matrices of left and right singular vectors respectively and Σ a diagonal matrix with singular values on the main diagonal. Capacity will be presented as:

(3.11) $$\begin{array}{r}\text{C}=E_H\{{\log }_2[det({\text{I}}_{\text{nR}}+\frac{\mathrm{\rho }}{\text{nT}}U\mathrm{\Sigma }{\mathrm{\Sigma }}^H{U}^H)]\}\text{bps}/\text{Hz}\end{array}$$

With the diagonalization of the product matrix, the channel capacity of MIMO is seen to include unitary and diagonal matrices only. Thus, total capacity of a MIMO channel consists of the sum of parallel AWGN SISO sub-channels. The number of parallel sub-channels is determined using the rank of the channel matrix. This rank of the channel matrix is given as:

(3.12) $$\begin{array}{r}\text{Rank}\left(\text{H}\right)=\text{k}\le min\{\text{nT},\text{ nR}\}\end{array}$$

Applying the rank (H) recalling that the determinant of a unitary matrix is equal to unity, the channel capacity can be represented respectively as:

(3.13) $$\begin{array}{r}C=E_H\{{\sum }_{i=1}^k{\log }_2(1+\frac{\rho }{nT}{\lambda }_i)\}\text{bps}/\text{Hz}\end{array}$$

(3.14) $$\begin{array}{r}=E_H\{{\sum }_{i=1}^k{\log }_2(1+\frac{\rho }{nT}{\sigma }_i^2)\}\text{bps}/\text{Hz}\end{array}$$

where ${\mathit{\lambda }}_{\mathit{i}}$, is the eigenvalues of the diagonal matrix ∧, and ${\mathit{\sigma }}_{\mathit{i}}^{\mathbf{\text{2}}}$, is the squares of the singular values of the diagonal matrix. The maximum capacity of MIMO is reached when n_T is received by an equal number of n_R antennas without interferences occurring, or where each transmitted signal is received by a separate set of receive antennas.

The capacity of the MIMO channel when channel information is unknown at the transmitter is maximized when all the eigenvalues ${⋋}_i={⋋}_j$ are equal. This happens when H is an orthogonal matrix [25]. When putting the constraints that each element of H has a power of unity. That is assumed to be equal, the maximized capacity is given as:

(3.15) $$\begin{array}{r}C=M{\log }_2(1+\frac{P_T}{N_0})\text{bps}/\text{Hz}\end{array}$$

This means that the capacity of an orthogonal MIMO channel is M times the capacity of a SISO channel.

Case 2:

When the channel matrix is understood at the transmitter; Here, the maximum capacity of the MIMO channel may be achieved by employing a technique of distributing power to a channel of lesser noise [26] on the transmit covariance matrix. Thus, the capacity is represented as:

(3.16) $$\begin{array}{r}C=E_H\left\{{\sum }_{i=1}^{k}lo{g}_2(1+{ϵ}_i\frac{\rho }{nT}{\lambda }_i)\right\}\end{array}$$

(3.17) $$\begin{array}{r}=E_H\left\{{\sum }_{i=1}^{k}lo{g}_2(1+{ϵ}_i\frac{\rho }{nT}{\sigma }_i^2)\right\}\text{bps}/\text{Hz}\end{array}$$

where ${ϵ}_i$ may be a scalar, representing the portion of the available transmit power going into the ith sub-channel. Power constraint at the transmitter can be represented as:

(3.18) $$\begin{array}{r}{\sum }_{i=1}^{nT}{ϵ}_i\le n_T.\end{array}$$

With a reduced range of non-zero singular values in the data rate equations, the capability of the MIMO channel is guaranteed to be reduced owing to a rank-deficient channel matrix. This situation occurs when the signals arriving at the receivers are correlated. Low correlation is not a guarantee of high capacity even though a high channel rank is necessary to obtain a high spectral efficiency in MIMO [26].

Adopted Model

The study adopted the water-filling model. This model is comparable to gushing water into a vessel. In communication systems, it’s accustomed to denote an inspiration in system design style and observe for exploit methods on communications channels. In analogous, Pascal law, the amplifies system in communications network repeaters or receivers amplifies every channel to the desired power level to compensate the channel impairments [26]–[30].

Water Filling Algorithm

The water-filing algorithm rule proves its optimality for channels having Additive White Gaussian Noise (AWGN) and inter-symbol interference (ISI). It is a customary baseline algorithmic rule for numerous digital communications systems. The entire quantity of water filled symbolizes the power allocated (Pt) to the system. See illustration in Fig. 4. This is given by

(3.19) $$\begin{array}{r}Totalpowerallocated=\frac{(P_t+\sum _{i=1}^n(1/H_i))}{\sum channel}-\frac{1}{H_i}\end{array}$$

where Pt is the power budget of MIMO system which is allocated among the different channels and H is the channel matrix of the system. Channel capability is given as:

(3.20) $$\begin{array}{r}capacity=\sum lo{g}_2(1+TotalpowerAlloc\ast H)\end{array}$$

Fig. 4. Water filling algorithm.

For the ergodic channel capacity, considering the two case situations, the (3.5) and (3.6) are to be applied severally.

The algorithms follow as below:

1. Re-order the MIMO sub-channel gain realization in an ascending order

2. Take the inverse of the channel gains.

3. Water filling has a non-uniform step structure due to the inverse of the channel gain.

4. ab initio, take the sum of the Total Power (Pt) and the Inverse of the channel gain. It provides the whole space in the water-filling and inverse power gain.

(3.21) $$\begin{array}{r}{P}_t+{\sum }_{i=1}^n\frac{1}{H}\end{array}$$

5. Decide the initial water level by the formula given below by taking the average power allocated (average water level)

(3.22) $$\begin{array}{r}\frac{P_t+\sum _{i=1}^n(1/H)}{\sum channels}\end{array}$$

6. The power values of each sub-channel are calculated by subtracting the inverse channel gain of each channel.

(3.23) $$\begin{array}{r}\text{Total }poweralloc=\frac{(P_t+\sum _{i=1}^n(1/H_i))}{\sum channel}-\frac{1}{H_i}\end{array}$$

7. Discontinue the iteration process at any level where the Power allocated value becomes negative.

Here Pt = total level of power provide,

$(1/H)$ = Noise level (ie inverse of channel gain).

Outage Capacity

Another index of system performances of the data rate (channel capacity) is the outage probability. Capacity on the channel instant response is supposed to remain constant throughout the transmission of a finite-length coded block of data. If it falls below the outage capacity, there is a chance that the transmit block of data may be decoded with error, whichever committal to the writing theme (coding scheme) is utilized. The chance that the capacity is a smaller amount than the outage capability is denoted by the letter q. This is expressed by

$$\begin{array}{r}Outagecapacity=C_{outage}\end{array}$$

(3.24) $$\begin{array}{r}q=Prob\{C\le C_{outage}\}\end{array}$$

In a finite probability (1 − q) where the channel capacity is higher than $C_{outage}$, it can also be represented as:

(3.25) $$\begin{array}{r}1-q=Prob\{C>C_{outage}\}\end{array}$$

Simulation Results and Discussion

During this section, the results are presented and discussed based on the outcome.

Parameters for Simulation

This study deploys several parameters to simulate the channel capacity (data rate) of wireless communication. These parameters were used to support the areas of consideration within the MATlab codes. Table I shows the assorted parameters, names, and values.

Table I. Parameters for Simulation

S/N	Parameter	Values
1	Transmitted power	[3], [2], [1], [4], [0]
2	Transmitted power	10
3	Capacity	4.7027
4	Total power	5
5	Input power	[4], [3], [2], [1], [0]
6	Noise power	[2], [3], [4], [1], [5]
7	Numbers of channels	5
8	Frequency	Rayliegh
9	Modulation	BPSK

Simulation Results

In Fig. 5, the simulation indicates that channel capacity increases as SNR increases. From the simulation output, the normal Shannon capacity is lower compared to when the antenna orientations (SIMO, SIMO, MISO and MIMO) simulated. A higher capacity was obtained when water filling was applied to MIMO. Fig. 6 compares the capacity of the channel for close loop state and at open loop state. Fig. 7 is the channel bit error probability graph. Fig. 8 outage probability graph. Fig. 9 is the curve of capacity (bps/hz) against SNR.

Fig. 5. Ergodic capacity of MIMO with water-filling.

Fig. 6. Ergodic capacity with and without CSIT at N_T = N_R = 4.

Fig. 7. BER for M IMO against noise power.

Fig. 8. Complementary CDF at SNR = 20 dB.

Fig. 9. Capacity gain of water filling MIMO.

Discussion

The MIMO channel capacity has so far been optimized based on the assumption that all transmit and receive antennas are used simultaneously. From the simulation results, there is a far difference in channel capacity when different arrays of antennas are used. In Fig. 5, and Fig. 6, channel improvement shown by the graph indicates that capacity increases as SNR increases. This correlates with [31], [32]. This was feasible when the spatial-multiplexing technique was suggested in an experiment [4]. Fig. 6 indicates that the closed-loop system optimizes more than the open-loop system at low SNR. It implies that at high SNR both cases have the same capacity despite CSI. We found that the relative performance of these techniques depends on the SNR and the relative values of N_T and N_R. The reason for the variation is that under a close loop, the transmit power is shared to a channel with less noise. This shows that a key part of the Eigen beamforming technique is choosing appropriate powers for each of the transmitted symbols. As SNR increases, there seems to be a convergence in capacity. However, the differences between the two conditions were slight. But there occurs a level shift ward in the graph showing lateral improvement in the channel. This correlate with several researches carried out by the majority referred [4], [5], [7], [13], [16], [18], [30], [31], [33]. These studies were based on variations in SNR in the systems. In their studies, techniques like; geometric programming, linear approximation, Legrand multipliers, singular value decomposition, and water-filling were used. Fig. 7, curves downward indicating that reduced bit error rate gave high performances on the channel. This high performance resulted from a rise condition of high SNR in the channel. With the water-filling technique employed in the channel, there is little variation in the channel performance at the cutoff point within the minimum and maximum values of the E_bN_o.

MIMO presents the best option in an outage condition. The SNR within the channel coverage areas is an optimum determinant for an outage. At reduced channel capacity, outage probability is extremely high (Fig. 8), therefore MIMO with water filling technique is most preferred. Complementary CDF is a better measurement tool to measure the performance of the channel. It enables network designers to foresee available bands, given the target data rate, depending on the quality of service (QoS) of the channels. The system capacity gain (Fig. 9), shows that gain increases steadily to a given SNR threshold and then starts decreasing steadily. However, without water filling, the gain decreases as SNR increases.

Conclusion

The performance of the wireless communication system is better when water water-filling algorithm is applied to MIMO. This technique allocates optimum power to sub-channels with less noise. These power allocation principles provide energy efficiency with respect to the corresponding average channel capacity by providing a high energy rate. The results of the simulations show that channel capacity is optimized by deploying the water-filling technique. The simulation indicates a linear increment in capacity when the system SNR increases. There was a significant performance of about 50% performance indication between conventional MIMO and MIMO with water-filling technique.